Employing baby universe surgery we study numerically the fractal structure of the intrinsic geometry of random surfaces coupled to matter field with c = 1. We simulate surfaces of the size up to 260.000 triangles. We observe the first indication that the branching and Haussdorff dimensions saturate. We estimate the branching dimension d(B) approximate to 2.6 and get a lower bound for the Hausdorff dimension d(H) > 3.0.